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Find the degree-2 Maclaurin polynomial of $f(x) = e^{ 8x}$ .
$f(0) = -6$ , $f'(0) = -6$ and $f''(0) = 4$ . Write the degree-2 Maclaurin polynomial of $f$ .
Find the degree-2 Taylor polynomial of $f(x) = 8\ln x$ centered at $x = 1$ .
On the interval, $|f^{(2)}(x)| \le 2$ . Bound the error of the degree-1 Taylor polynomial centered at $a$ , at a point with $|x - a| = \tfrac{1}{2}$ .
$|f^{(3)}(x)| \le 9$ on $[0, 1]$ . Bound the error of the degree-2 Maclaurin polynomial at $x = 1$ .
The Lagrange error bound for the degree-4 Taylor polynomial uses the maximum of which derivative?
1. $f^{(5)}$ , the next derivative
2. $f^{(4)}$
3. $f$ itself
The Lagrange error bound for the degree-6 Taylor polynomial uses the maximum of which derivative?
1. $f^{(7)}$ , the next derivative
2. $f^{(6)}$
3. $f$ itself
The Lagrange error bound for the degree-5 Taylor polynomial uses the maximum of which derivative?
1. $f^{(6)}$ , the next derivative
2. $f^{(5)}$
3. $f$ itself
The Lagrange error bound for the degree-3 Taylor polynomial uses the maximum of which derivative?
1. $f^{(4)}$ , the next derivative
2. $f^{(3)}$
3. $f$ itself
The Lagrange error bound for the degree-2 Taylor polynomial uses the maximum of which derivative?
1. $f^{(3)}$ , the next derivative
2. $f^{(2)}$
3. $f$ itself
Find the radius of convergence of $\displaystyle\sum_{n=0}^{\infty} \dfrac{x^n}{ 8^n}$ .
What is the interval of convergence of $\displaystyle\sum_{n=1}^{\infty} \dfrac{(x - 7)^n}{n}$ ?
1. $[6, 8)$
2. $(6, 8)$
3. $[6, 8]$
What is the interval of convergence of $\displaystyle\sum_{n=1}^{\infty} \dfrac{(x - 4)^n}{n}$ ?
1. $[3, 5)$
2. $(3, 5)$
3. $[3, 5]$
What is the interval of convergence of $\displaystyle\sum_{n=1}^{\infty} \dfrac{(x - 8)^n}{n}$ ?
1. $[7, 9)$
2. $(7, 9)$
3. $[7, 9]$
What is the interval of convergence of $\displaystyle\sum_{n=1}^{\infty} \dfrac{(x - 3)^n}{n}$ ?
1. $[2, 4)$
2. $(2, 4)$
3. $[2, 4]$
What is the interval of convergence of $\displaystyle\sum_{n=1}^{\infty} \dfrac{(x - 1)^n}{n}$ ?
1. $[0, 2)$
2. $(0, 2)$
3. $[0, 2]$
What is the interval of convergence of $\displaystyle\sum_{n=1}^{\infty} \dfrac{(x - 9)^n}{n}$ ?
1. $[8, 10)$
2. $(8, 10)$
3. $[8, 10]$
What is the interval of convergence of $\displaystyle\sum_{n=1}^{\infty} \dfrac{(x - 5)^n}{n}$ ?
1. $[4, 6)$
2. $(4, 6)$
3. $[4, 6]$
What is the interval of convergence of $\displaystyle\sum_{n=1}^{\infty} \dfrac{(x - 6)^n}{n}$ ?
1. $[5, 7)$
2. $(5, 7)$
3. $[5, 7]$
What is the interval of convergence of $\displaystyle\sum_{n=1}^{\infty} \dfrac{(x - 2)^n}{n}$ ?
1. $[1, 3)$
2. $(1, 3)$
3. $[1, 3]$
Find the radius of convergence of $\displaystyle\sum_{n=0}^{\infty} (2x)^n$ .
Find the coefficient of $x^3$ in the Maclaurin series of $e^{ 5x}$ .
$\displaystyle\sum_{n=0}^{\infty} \dfrac{(-1)^n (1x)^{2n+1} }{(2n + 1)!}$ is the Maclaurin series of which function?
1. $\sin(1x)$
2. $\cos(1x)$
3. $e^{ 1x}$
$\displaystyle\sum_{n=0}^{\infty} \dfrac{(-1)^n (2x)^{2n+1} }{(2n + 1)!}$ is the Maclaurin series of which function?
1. $\sin(2x)$
2. $\cos(2x)$
3. $e^{ 2x}$
$\displaystyle\sum_{n=0}^{\infty} \dfrac{(-1)^n (6x)^{2n+1} }{(2n + 1)!}$ is the Maclaurin series of which function?
1. $\sin(6x)$
2. $\cos(6x)$
3. $e^{ 6x}$
$\displaystyle\sum_{n=0}^{\infty} \dfrac{(-1)^n (8x)^{2n+1} }{(2n + 1)!}$ is the Maclaurin series of which function?
1. $\sin(8x)$
2. $\cos(8x)$
3. $e^{ 8x}$
$\displaystyle\sum_{n=0}^{\infty} \dfrac{(-1)^n (7x)^{2n+1} }{(2n + 1)!}$ is the Maclaurin series of which function?
1. $\sin(7x)$
2. $\cos(7x)$
3. $e^{ 7x}$
$\displaystyle\sum_{n=0}^{\infty} \dfrac{(-1)^n (9x)^{2n+1} }{(2n + 1)!}$ is the Maclaurin series of which function?
1. $\sin(9x)$
2. $\cos(9x)$
3. $e^{ 9x}$
$\displaystyle\sum_{n=0}^{\infty} \dfrac{(-1)^n (5x)^{2n+1} }{(2n + 1)!}$ is the Maclaurin series of which function?
1. $\sin(5x)$
2. $\cos(5x)$
3. $e^{ 5x}$
$\displaystyle\sum_{n=0}^{\infty} \dfrac{(-1)^n (3x)^{2n+1} }{(2n + 1)!}$ is the Maclaurin series of which function?
1. $\sin(3x)$
2. $\cos(3x)$
3. $e^{ 3x}$
$\displaystyle\sum_{n=0}^{\infty} \dfrac{(-1)^n (4x)^{2n+1} }{(2n + 1)!}$ is the Maclaurin series of which function?
1. $\sin(4x)$
2. $\cos(4x)$
3. $e^{ 4x}$
Find the coefficient of $x^4$ in the Maclaurin series of $\cos(2x)$ .
Find the coefficient of $x^3$ in the Maclaurin series of $xe^{ 4x}$ .
Find the coefficient of $x^{ 10 }$ in the Maclaurin series of $\dfrac{1}{1 + x^2}$ .
$\arctan x = \displaystyle\int_0^x \dfrac{dt}{1 + t^2}$ . Find the coefficient of $x^{ 3 }$ in its Maclaurin series.

Answers

$32 x^{2} + 8 x + 1$
$2 x^{2} - 6 x - 6$
$8 x - 4 \left(x - 1\right)^{2} - 8$
$\frac{1}{4}$
$\frac{3}{2}$
$f^{(5)}$ , the next derivative
$8$
$[6, 8)$
$\frac{1}{2}$
$\frac{125}{6}$
$\sin(1x)$
$\frac{2}{3}$
$8$
$-1$
$- \frac{1}{3}$